Homogenization method for strength and inelastic behavior of nanocrystalline materials

Homogenization techniques are used for modeling the so-called “breakdown” of the Hall–Petch law in the case of nanocrystalline (NC) materials. In this paper, the NC material is modeled as a composite material composed of two phases: the grain core (inclusion) and the grain boundaries (matrix). The deformation of the inclusion phase has a viscoplastic component that takes into account the dislocation glide mechanism as well as Coble creep. The boundary phase is modeled as an amorphous material with a perfect elastic–plastic behavior. An application of the model is developed on pure copper under tensile load. The results are compared with various experimental data.     

On the elastic–viscoplastic behavior of nanocrystalline materials

A new constitutive law is introduced to quantify the macroscopic effect of grain boundary dislocation emission on the behavior of pure face center cubic nanocrystalline materials. It is postulated that an emitted dislocation ends its trajectory in the grain boundary opposite to the source causing mass transfer. Dislocation emission by grain boundary ledges, considered here as the primary grain-boundary sources, is modeled as a thermally activated mechanism and the penetration of an emitted dislocation is assimilated as a soft collision. The macroscopic behavior of the material is retrieved via the use of a secant self-consistent scheme. The material is seen as a two-phase composite where the inclusion phase represents grain cores, their behavior is driven by dislocation glide, and where the matrix phase, governed by the newly introduced dislocation emission and penetration mechanism, represents both grain boundaries and triple junctions. The long range stress field arising from the presence of grain boundaries is taken into account in the critical glide resistance stress at 0 K in the inclusion phase. The model is applied to polycrystal copper and results in pure tension and creep are compared to experiments. Good agreements between the experimental measurements and the model predictions are observed.     

A variational multiscale constitutive model for nanocrystalline materials

This paper presents a variational multi-scale constitutive model in the finite deformation regime capable of capturing the mechanical behavior of nanocrystalline (nc) fcc metals. The nc-material is modeled as a two-phase material consisting of a grain interior phase and a grain boundary effected zone (GBAZ). A rate-independent isotropic porous plasticity model is employed to describe the GBAZ, whereas a crystal-plasticity model which accounts for the transition from partial dislocation to full dislocation mediated plasticity is employed for the grain interior. The constitutive models of both phases are formulated in a small strain framework and extended to finite deformation by use of logarithmic and exponential mappings. Assuming the rule of mixtures, the overall behavior of a given grain is obtained via volume averaging. The scale transition from a single grain to a polycrystal is achieved by Taylor-type homogenization where a log-normal grain size distribution is assumed. It is shown that the proposed model is able to capture the inverse Hall-Petch effect, i.e., loss of strength with grain size refinement. Finally, the predictive capability of the model is validated against experimental results on nanocrystalline copper and nickel.     

微纳米晶金属的应变率敏感性及应变硬化行为分析

基于亚微米、纳米晶粒组织塑性变形过程中多种变形机制(位错机制、扩散机制及晶界滑动机制)共存,建立了理论模型,用于定量研究亚微米、纳米晶粒组织的塑性变形行为.以铜为模型材料,计算分析了晶粒尺度、应变率以及温度对亚微米、纳米晶粒组织塑性变形行为的影响.结果表明:相比粗晶铜,亚微米晶铜表现出明显的应变率敏感性,并且应变率敏感系数随晶粒尺度及变形速率的减小而增大;同时,增大变形速率或降低变形温度都能提高材料的应变硬化能力,延缓颈缩发生,进而提高材料的延性.计算分析结果与实验报道吻合.     

Brittle versus ductile transition of nanocrystalline metals

The brittle versus ductile transition for conventional metals is dictated by the competition between dislocation emission and cleavage. For nanocrystalline metals with grain size below 25 nm, however, dislocation activities are suppressed and the classic theory fails to apply. In this paper, one of the competing mechanisms that control the brittle versus ductile transition of nanocrystalline metals is found to be the grain boundary dominated creep deformation versus the grain boundary decohesion. A model is proposed to quantify the crack propagation in nanocrystalline metals. The effects of material properties, initial configuration and applied loads on the property of crack propagation are addressed. It is concluded that either the increases in the initial crack length, the applied load and the grain boundary damage, or the deterrence in creep deformation, accelerate the crack propagation, and vice versa.     

Investigating the deformation of nanocrystalline copper with microscale kinematic metrics and molecular dynamics

Atomistic simulations are employed to investigate the deformation of nanocrystalline copper and the associated strain accommodation mechanisms at 10 K as a function of grain size. Volume-averaged kinematic metrics based on continuum mechanics theory are formulated to analyze the results of molecular dynamics simulations. The metrics rely on both reference and current configurations, along with nearest neighbor lists to estimate nanoscale behavior of atomic deformation fields in nanocrystalline copper. Various deformation mechanisms are activated in the structures, and shown to depend on average grain size of the nanocrystalline structure. Furthermore, grain boundaries, along with dislocation glide, become an important source of strain accommodation as grain size is reduced. It is demonstrated that the metrics capture the contributions of various mechanisms, and provide a sense of the history of atomic regions undergoing both elastic and plastic deformation. The significance of this research is that unique kinematic signatures of the mechanisms are uncovered using certain metrics, and we are able to resolve the contributions of the deformation mechanisms to the overall strain of the structure using Green strain.     

Activation volume and deviation from Cottrell–Stokes law at small grain size

Dependence of activation volume with flow stress is examined for metals with grain size lower than 0.3 μm and larger than few tens of nanometers, where plastic deformation is most likely to be governed by a combination of grain boundary sliding and dislocations activity. The experimentally observed deviation from the classic linear behavior given by Cottrell–Stokes law [Basinski, Z.S., 1974. Forest hardening in face centered cubic metals. Scripta Metallurgica 8, 1301–1308] is analyzed, thanks to a modified Orowan equation taking into account of the grain boundaries sliding coupled to dislocations activity. These results are compared to experimental measurements of the activation volume, between room temperature and 120 °C, for a copper nanostructure with a grain size of 100 nm. A constant activation volume is observed at low stress (or high temperature) followed by an increase of activation volume with stress (inverse Cottrell–Stokes behavior). This analysis follows our initial experiments on this fine grained metal prepared by powder metallurgy, which exhibits ductility at near constant stress and strain rate [Champion, Y., Langlois, C., Guérin-Mailly, S., Langlois, P., Bonnentien, J.-L., Hÿtch, M.J., 2003. Near-perfect elastoplasticity in pure nanocrystalline copper. Science 300, 310–311].     

Crack growth versus blunting in nanocrystalline metals with extremely small grain size

Fracture of nanocrystalline metals with extremely small grain size is simulated in this paper by structural evolution. Two-dimensional scheme is formulated to study the competition between crack growth and blunting in nanocrystalline samples with edge cracks. The scheme couples the creep deformation induced by grain boundary (GB) mechanisms and the intergranular crack growth. The effects of material properties, initial configurations and applied loads are explored. Either the enhancement in diffusion mobility, or the deterrence in the grain boundary damage, would blunt the crack and decelerate its growth, and vice versa. The simulations agree with the analytical predictions as modified from that of Yang and Yang [2008. Brittle versus ductile transition of nanocrystalline metal. Int. J. Solids Struct. 45, 3897-3907]. Upon the suppression of dislocation activities, it is validated that the brittle versus ductile transition of nanocrystals is controlled by the development of grain boundary-dominated creep versus grain boundary decohesion. Further simulations found that either decreasing the grain sizes or increasing the dispersion of grain sizes would blunt the growing cracks.     

THE RATE-INDEPENDENT CONSTITUTIVE MODELING FOR POROUS AND MULTI-PHASE NANOCRYSTALLINE MATERIALS

To determine the time-independent constitutive modeling for porous and multiphase nanocrystalline materials and understand the effects of grain size and porosity on their mechanical behavior, each phase was treated as a mixture of grain interior and grain boundary, and pores were taken as a single phase, then Budiansky's self-consistent method was used to calculate the Young's modulus of porous, possible multi-phase, nanocrystalline materials, the prediction being in good agreement with the results in the literature. Further, the established method is extended to simulate the constitutive relations of porous and possible multi-phase nanocrystalline materials with small plastic deformation in conjunction with the secant-moduli approach and iso-strain assumption. Comparisons between the experimental grain size and porosity dependent mechanical data and the corresponding predictions using the established model show that it appears to be capable of describing the time-independent mechanical behaviors for porous and multi-phase nanocrystalline materials in a small plastic strain range. Further discussion on the modification factor, the advantages and limitations of the model developed were present.     

THE RATE-INDEPENDENT CONSTITUTIVE MODELING FOR POROUS AND MULTI-PHASE NANOCRYSTALLINE MATERIAL

To determine the time-independent constitutive modeling for porous and multiphase nanocrystalline materials and understand the effects of grain size and porosity on their mechanical behavior, each phase was treated as a mixture of grain interior and grain boundary, and pores were taken as a single phase, then Budiansky's self-consistent method was used to calculate the Young's modulus of porous, possible multi-phase, nanocrystalline materials, the prediction being in good agreement with the results in the literature. Further, the established method is extended tosimulate the constitutive relations of porous and possible multi-phase nanocrystalline materials with small plastic deformation in conjunction with the secant-moduli approach and iso-strain assumption. Comparisons between the experimental grain size and porosity dependent mechanical data and the corresponding predictions using the established model show that it appears to be capable of describing the time-independent mechanical behaviors for porous and multi-phase nanocrystalline materials in a small plastic strain range. Further discussion on the modification factor, the advantages and limitations of the model developed were present.     

铜双晶体的循环应力-应变响应

比较了铜单晶体和多晶体疲劳行为的异同,提出了研究双晶体疲劳行为的必要性．总结了具有不同晶体取向和晶界的铜双晶体的疲劳行为的最新进展．利用平行晶界铜双晶体的取向因子和晶界影响区,总结了在循环载荷作用下的晶界强化模型．分析了垂直晶界铜双晶体循环塑性变形行为的特点,讨论了组元晶体取向对垂直晶界铜双晶体循环应力－应变曲线的影响．提出了提高单晶体和双晶体疲劳强度的控制因素．     

Constitutive modeling for nanocrystalline metals based on cooperative grain boundary mechanisms

In nanocrystalline metals, the plastic deformation is accommodated primarily at the grain boundaries. Yang and Wang (J. Mech. Phys. Solids, 2003) suggested a deformation model based on clusters consisting of nine grains and incorporating both the Ashby-Verrall mechanism and a 30° rotation of closely linked pairs of grains. In the present article, the insertion and rotation processes are considered together as a cooperative deformation mechanisms, and the degree to which each process contributes is determined by the application of the principle of maximum plastic work. Plane strain and three-dimensional constitutive relations based on this concept are derived for which a general stress state drives the orientation evolution of various grain clusters under the Reuss assumption. Detailed calculation shows that the strain rate depends linearly on the stress, with the values of the coefficients in this linear relationship dictated by the microscopic energy dissipation. The deformation contributed to the overall response by the grain boundary mechanism is discussed in the spirit of the Hashin-Shtrikman bounds.     

Mechanical response and modeling of fully compacted nanocrystalline iron and copper

A comprehensive study on the response of nanocrystalline iron and copper to quasi-static and dynamic loading is reported. Bulk solid nanocrystalline iron and copper specimens used in static and dynamic loading experiments were made by compaction and hot sintering of the nanocrystalline powders. The powders, with grain size 16–96 nm, were obtained by using high energy ball milling. The stress/strain response of dense nanocrystalline iron is found to be grain size and strain rate dependent. The KHL model is modified by incorporating Hall–Petch relation (i.e. yield stress dependence on grain size) and is used to represent the behavior of fully compacted nanocrystalline material. A good correlation with the experimental results is demonstrated.     

Constitutive model for plastic deformation of nanocrystalline materials with shear band

A phase mixture model was used to study the plastic deformation behaviors in hardening stage of nanocrystalline materials. The material was considered as a composite of grain interior phase and grain boundary (GB) phase. The constitutive equations of the two phases were determined in term of their main deformation mechanisms. In softening stage, a shear band deformation mechanism was presented and the corresponding constitutive relation was established. Numerical simulations have shown that the predications fit well with experimental data. The investigation using the finite-element method (FEM) provided a direct insight into quantifying shear localization effect in nanocrystalline materials.     

Mechanics modeling for deformation of nano-grained metals

The electro-deposition technique is capable of producing nano-grained bulk copper specimens that exhibit superplastic extensibility at room temperature. Metals of such small grain sizes deform by grains squeezing past each other, with little distortion occurring in the grain cores. Accommodation mechanisms such as grain boundary diffusion and grain rotation control the kinetics of the process. A model of a 9-grain cluster is proposed that incorporates both the Ashby-Verrall mechanism and the 30° rotation of closely linked grain pairs. A constitutive relation is derived that relates the creep strain rate linearly to the difference between the applied stress and a threshold stress. The creep rate and the threshold stress predicted by the model are in quantitative agreement with the experimental data.     

Dislocation nucleation from bicrystal interfaces and grain boundary ledges: Relationship to nanocrystalline deformation

Molecular dynamics simulations are used to evaluate the primary interface dislocation sources and to estimate both the free enthalpy of activation and the critical emission stress associated with the interfacial dislocation emission mechanism. Simulations are performed on copper to study tensile failure of a planar Σ5 {2 1 0} 53.1° interface and an interface with the same misorientation that contains a ledge. Simulations reveal that grain boundary ledges are more favorable as dislocation sources than planar regions of the interface and that their role is not limited to that of simple dislocation donors. The parameters extracted from the simulations are utilized in a two-phase composite mesoscopic model for nanocrystalline deformation that includes the effects of both dislocation emission and dislocation absorption mechanisms. A self-consistent approach based on the Eshelby solution for grains as ellipsoidal inclusions is augmented by introduction of stress concentration in the constitutive law of the matrix phase to account for more realistic grain boundary effects. Model simulations suggest that stress concentration is required in the standard continuum theory to activate the coupled grain boundary dislocation emission and absorption mechanisms when activation energy of the dislocation source is determined from atomistic calculation on grain boundaries without consideration of impurities or other extrinsic defects.     

Experimental Investigation of Plastic Strain Recovery and Creep in Nanocrystalline Copper Thin Films

Nanocrystalline metals exhibit a phenomenon called plastic strain recovery whereby plastic strain introduced through a load cycle is gradually recovered under no external loading over a time period of hours and days. In this study, we experimentally explore the diffusive mechanisms and the strain rates for nanocrystalline thin films of copper with an average grain size of about 35 nm during plastic strain recovery and creep. The experiments are performed via the plane strain bulge test and the thin film samples are deposited using thermal evaporation and sputtering. The specimens recover their residual strain in a period of time with two characteristic strain rates, a transient strain recovery rate of the order of 10^?7/s and a steady-state strain recovery rate of the order of 10^?9/s and there is a characteristic time at which the transition occurs between the two rates. The results suggest that a diffusive mechanism in conjunction with voids within the nanocrystalline material can explain the two plastic strain recovery rates and the transition between the two.     

The competition of grain size and porosity in the viscoplastic response of nanocrystalline solids

Many important processing techniques for nanocrystalline solids, such as ball milling and compaction, are frequently accompanied by the presence of voids in the end products. These voids can apparently lower the yield strength of the material. In order to address the issue of competition between grain size and porosity, we develop an explicit, analytical composite model that allows us to determine the viscoplastic response of a porous, nanocrystalline solid. The development made use of the concept of a three-phase composite comprising of the plastically harder grain interior, plastically softer grain-boundary affected zone (GBAZ), and porosity. A homogenization theory that accounts for the evolution of porosity during plastic flow is established. This establishment is built upon the extension of a linear viscoelastic composite to a non-linear viscoplastic one, in which the viscoplastic behavior of the constituent phases is represented by a unified constitutive law. Then by means of a field fluctuation method, the local strain rates are linked to the applied total strain rate. Such a linkage in turn provides the secant viscosity of the constituent phases at every stage of deformation. In order to test the applicability of the developed theory, we have applied it to model the viscoplastic response of an iron and an iron–copper mixture tested by Khan et al. [Khan, A.S., Zhang, H., Takacs, L., 2000. Mechanical response and modeling of fully compacted nanocrystalline iron and copper. Int. J. Plasticity 16, 1459–1476] and Khan and Zhang [Khan, A.S., Zhang, H., 2000. Mechanically alloyed nanocrystalline iron and copper mixture: behavior and constitutive modeling over a wide range of strain rates. Int. J. Plasticity 16, 1477–1492]. It is demonstrated that the theory is capable of capturing the major features of the tested results at various grain sizes and porosities. Our calculations further point to the change of yield strength in the Hall–Petch plot from an initial increase to level off, and then to decline, at various porosities under a constant strain-rate loading. This in turn brings about the existence of a critical grain size in the nano-meter range at which the material exhibits maximum yield strength. Moreover, this critical grain size tends to move to the left in the Hall–Petch plot as the GBAZ becomes softer.     

A Micromechanics Study of Lamellar TiAl

Microdeformation patterns of lamellar TiAl specimens with various grain sizes under uniaxial tension are mapped using the micro/nano experimental mechanics technique called SIEM (Speckle Interferometry w ith Electron Microscopy). The stress–strain relationships were obtained from deformations within decreasing areas ranging from mm² to μm². We found that the stress–strain relationship of the material depends on the size of strain measuring area in relation to the grain size. The stiffness at a grain boundary can be as large as 7–10 times more than that of the grain itself. From the data obtained so far, it seems that the traditional way of using PST (polysynthetically twinned) single crystal to predict polycrystalline behavior may not be appropriate.     

考虑尺寸效应的多晶金属循环塑性分析

本文采用多晶塑性分析方法,设材料点包含一定数量的各向异性单晶晶粒并考虑晶粒尺寸的影响,计算材料点的应力和应变时利用了Taylor假设。模型引入考虑尺寸效应的晶体滑移硬化函数,同时针对晶体滑移引入背应力及其方向性硬化的描述,以反映不同晶粒尺寸材料在循环加载条件下的力学行为。利用该模型,本文第一作者采用显式格式编制了与ABAQUS商用有限元软件接口的用户材料子程序（VUMAT）,实例计算证实该模型可以反映和描述多晶金属材料在材料反复加载条件下的循环塑性行为与尺寸效应。